[Paper Review] Large deviations for a random speed particle

R. Lefevere, Mauro Mariani+1 more|arXiv (Cornell University)|Mar 14, 2011

Stochastic processes and statistical mechanics10 references3 citations

TL;DR

This paper investigates large deviations for the empirical measure of a particle moving in a box with stochastic speed resets at boundaries, where speeds are i.i.d. from a heavy-tailed distribution. It shows that when the tail of the speed distribution is sufficiently oscillating, the empirical measure fails to satisfy a standard large deviations principle, and instead exhibits optimal lower and upper rate functionals that differ significantly from the Donsker-Varadhan functional due to long-range dependence induced by heavy tails.

ABSTRACT

We investigate large deviations for the empirical measure of the position and momentum of a particle traveling in a box with hot walls. The particle travels with uniform speed from left to right, until it hits the right boundary. Then it is absorbed and re-emitted from the left boundary with a new random speed, taken from an i.i.d. sequence. It turns out that this simple model, often used to simulate a heat bath, displays unusually complex large deviations features, that we explain in detail. In particular, if the tail of the update distribution of the speed is sufficiently oscillating, then the empirical measure does not satisfy a large deviations principle, and we exhibit optimal lower and upper large deviations functionals.

Motivation & Objective

• To analyze large deviations of the empirical measure for a particle with random-speed renewal dynamics in a bounded interval.
• To identify conditions under which the standard Donsker-Varadhan large deviations principle fails to hold.
• To characterize the behavior of the empirical measure when the speed distribution has heavy tails that induce long return times.
• To construct optimal lower and upper large deviations functionals in cases where the standard rate functional is invalid.
• To explain why standard renewal theory and inversion techniques fail in the presence of heavy-tailed return times.

Proposed method

• Model the particle as a Markov process with i.i.d. speed updates upon hitting the right boundary, using a renewal process structure with inter-collision times $\tau_i = 1/v_i$.
• Define the empirical measure $\mu_t = \frac{1}{t} \int_0^t \delta_{(q_s,p_s)} \, ds$ on $[0,1) \times \mathbb{R}_+$, and study its large deviations under the weak topology.
• Use the generator $L$ of the process, given by $Lg(q,p) = p \partial_q g$ with a boundary condition $g(1,p) = \int \psi(d\tau) g(0,1/\tau)$, to analyze the dynamics.
• Apply the Donsker-Varadhan variational formula $I(\mu) = \sup_{g>0, g \in D(L)} (-\langle g^{-1} Lg \rangle_\mu)$ to compute the rate functional.
• Analyze the behavior of $\mathbb{E}[\exp(c/v_i)]$ for $c>0$ to determine whether the tail of the speed distribution leads to non-negligible probabilities of very slow speeds.
• Construct optimal lower and upper large deviations functionals when the standard Donsker-Varadhan functional fails due to oscillating tails in the speed distribution.

Experimental results

Research questions

• RQ1 Under what conditions on the speed distribution does the empirical measure of the random-speed particle fail to satisfy a large deviations principle?
• RQ2 How do heavy-tailed distributions of the speed $v_i$ affect the convergence to the invariant measure and the validity of standard large deviations theory?
• RQ3 What is the structure of the rate functional when the Donsker-Varadhan functional is not applicable due to long return times?
• RQ4 Why do standard renewal techniques and inversion methods fail in this model when the return times $1/v_i$ have heavy tails?
• RQ5 Can optimal lower and upper large deviations functionals be constructed in cases where the standard LDP does not hold?

Key findings

• The empirical measure does not satisfy a large deviations principle if the tail of the speed distribution is sufficiently oscillating and $\mathbb{E}[\exp(c/v_i)] = \infty$ for some $c>0$.
• When the Donsker-Varadhan functional fails, the rate functional can differ significantly from the standard form, with the empirical measure potentially concentrating on measures singular with respect to the invariant measure.
• The presence of heavy tails in the speed distribution leads to a slow convergence to the invariant measure and invalidates standard large deviations techniques based on exponential moments.
• The failure of standard methods is due to the lack of regularity in the renewal process and the unboundedness of the inverse of the speed, which disrupts the applicability of the inversion map.
• Optimal lower and upper large deviations functionals are constructed explicitly in the case of oscillating tails, showing that the system exhibits non-standard rare event behavior.
• The analysis reveals that events of exponentially small probability can be dominated by the occurrence of very slow velocities $v_i \sim t^{-1}$, which are not negligible at the large deviations scale.